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Category Archives: proof
Efficiency of repeated squaring: another proof
In my previous post I proved that the “binary algorithm” (corresponding to the binary expansion of a number ) is the most efficient way to build using only doubling and incrementing steps. Today I want to explain another nice proof, … Continue reading
Posted in computation, proof
Tagged binary, double, efficiency, exponent, increment, proof, repeated, squaring, steps
3 Comments
Efficiency of repeated squaring: proof
My last post proposed a claim: The binary algorithm is the most efficient way to build using only doubling and incrementing steps. That is, any other way to build by doubling and incrementing uses an equal or greater number of … Continue reading
Posted in computation, proof
Tagged binary, double, efficiency, exponent, increment, proof, repeated, squaring, steps
2 Comments
The chromatic number of the plane, part 4: an upper bound
In my previous posts I explained lower bounds for the Hadwiger-Nelson problem: we know that the chromatic number of the plane is at least 5 because there exist unit distance graphs which we know need at least 5 colors. Someday, … Continue reading
Iterating squared digit sums in other bases
In a previous post I wrote about iterating the squared digit sum function, which adds up the sum of the squares of the digits of a number; for example, . Denis left a comment asking about other bases—what happens if … Continue reading
The chromatic number of the plane, part 3: a new lower bound
In my previous post I explained how we know that the chromatic number of the plane is at least 4. If we can construct a unit distance graph (a graph whose edges all have length ) which needs at least … Continue reading
The chromatic number of the plane, part 2: lower bounds
In a previous post I explained the Hadwiger-Nelson problem—to determine the chromatic number of the plane—and I claimed that we now know the answer is either 5, 6, or 7. In the following few posts I want to explain how … Continue reading
Iterating squared digit sum
Another fun fact I learned from John Cook. Let be the function which takes a positive integer and outputs the sum of the squares of its digits. For example, . Since the output is itself another positive integer, we can … Continue reading
Posted in arithmetic, computation, proof
Tagged digit, loop, Porges, proof, squared, sum
12 Comments
The chromatic number of the plane, part 1
About a week ago, Aubrey de Grey published a paper titled “The chromatic number of the plane is at least 5”, which is a really cool result. It’s been widely reported already, so I’m actually a bit late to the … Continue reading
Properties of orthogons I
First things first: from now on, when talking about polygons with only right angles, instead of calling them “orthogonal polygons” I’m going to start calling them “orthogons”, which sounds cool, is much less clunky than “orthogonal polygons”, and doesn’t seem … Continue reading
Posted in combinatorics, geometry, proof
Tagged concave, convex, orthogonal, orthogons, polygons, proof, properties, vertices
10 Comments
Fermat’s Little Theorem: proof by group theory
It’s time for our third and final proof of Fermat’s Little Theorem, this time using some group theory. This proof is probably the shortest—explaining this proof to a professional mathematician would probably take only a single sentence—but requires you to … Continue reading
Posted in group theory, number theory, primes, proof
Tagged combinatorics, group, order, proof, theory
Comments Off on Fermat’s Little Theorem: proof by group theory